Binary Operations Properties – SS3 Mathematics Lesson Note (First Term)

A binary operation is a rule for combining any two elements of a given non-empty set. Combining two elements may give another element, which may or may not belong to the set under consideration. A binary operation is usually denoted by symbols such as * and ⊕.

Behavioural objectives

By the end of the lesson, students should be able to:

• define a binary operation;
• test whether a set is closed under a binary operation;
• show whether a binary operation is commutative;
• show whether a binary operation is associative;
• show whether one binary operation distributes over another.

What are the properties of a binary operation?

Closure property

A non-empty set Z is closed under a binary operation * if a * b ∈ Z for all a, b ∈ Z.

Example: A binary operation * is defined on the set S = {0, 1, 2, 3, 4} by x * y = x + y – xy. Find:

1. 2 * 4
2. 3 * 1
3. 0 * 3

Is the set S closed under the operation *?

Solution

1. 2 * 4, that is x = 2, y = 4: 2 + 4 – (2 × 4) = 6 – 8 = -2
2. 3 * 1 = 3 + 1 – (3 × 1) = 4 – 3 = 1
3. 0 * 3 = 0 + 3 – (0 × 3) = 3

Since -2 ∉ S, the operation * is not closed in S.

Commutative property

If S is a non-empty set closed under the binary operation *, then the operation * is commutative if a * b = b * a for all a, b ∈ S.

Therefore, a binary operation is commutative if the order of combination does not affect the result.

Example: The operation * on the set R of real numbers is defined by p * q = p³ + q³ – 3pq. Is the operation commutative?

Solution

p * q = p³ + q³ – 3pq

Commutative condition: p * q = q * p

To obtain q * p, use the same operation as p * q, but replace p by q and q by p.

Hence, q * p = q³ + p³ – 3qp

Since q³ + p³ – 3qp = p³ + q³ – 3pq, we have p * q = q * p. The operation is commutative.

Associative property

If a non-empty set S is closed under a binary operation *, that is a * b ∈ S, then the operation is associative if (a * b) * c = a * (b * c) for all a, b, c ∈ S.

Example: The operation ⊕ on the set Z of integers is defined by a ⊕ b = 2a + 3b – 1. Determine whether or not the operation is associative in Z.

Solution

Introduce another element c.

Associative condition: (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c)

LHS: (a ⊕ b) ⊕ c = (2a + 3b – 1) ⊕ c

= 2(2a + 3b – 1) + 3c – 1

= 4a + 6b – 2 + 3c – 1

= 4a + 6b + 3c – 3

RHS: a ⊕ (b ⊕ c) = a ⊕ (2b + 3c – 1)

= 2a + 3(2b + 3c – 1) – 1

= 2a + 6b + 9c – 3 – 1

= 2a + 6b + 9c – 4

Since (a ⊕ b) ⊕ c ≠ a ⊕ (b ⊕ c), the operation is not associative in Z.

Distributive property

If a set S is closed under two binary operations * and ∘, then * is distributive over ∘ if, for all a, b, c ∈ S:

• a * (b ∘ c) = (a * b) ∘ (a * c) (left distributive)
• (b ∘ c) * a = (b * a) ∘ (c * a) (right distributive)

Example: Given the set R of real numbers under the operations * and ∘ defined by a * b = a + b – 3 and a ∘ b = 5ab for all a, b ∈ R, does * distribute over ∘?

Solution

Let a, b, c ∈ R.

The condition is a * (b ∘ c) = (a * b) ∘ (a * c).

a * (b ∘ c) = a * (5bc)

= a + 5bc – 3

(a * b) ∘ (a * c) = (a + b – 3) ∘ (a + c – 3)

= 5(a + b – 3)(a + c – 3)

From the expansion, it is clear that a * (b ∘ c) ≠ (a * b) ∘ (a * c). Therefore * does not distribute over ∘.

Evaluation

Closure

An operation * defined on the set R of real numbers is x * y = 3x + 2y – 1, x, y ∈ R.

1. Determine (a) 2 * 3 (b) -4 * 5 (c) 1 * 1
2. Is the operation closed?

Commutative and associative properties

1. A binary operation * is defined on the set R of real numbers by x * y = x + y + 3xy for all x, y ∈ R. Determine whether or not * is: (a) commutative, (b) associative.
2. The operation ⊕ on the set R of real numbers is defined by a ⊕ b = (a + b)/2 + ab for a, b ∈ R. Show that the operation ⊕ is commutative but not associative on R.

General evaluation

1. The operation * on the set R of real numbers is defined by x * y = 3x + 2y – 1, x, y ∈ R. Determine (i) 2 * 3 (ii) 1/3 * 1/2 (iii) -4 * 5
2. The operation * on the set R of real numbers is defined by p * q = p³ + q³ – 3pq; p, q ∈ R. Is the operation * commutative in R?
3. The operations * and ⊕ are defined on the set R of natural numbers by a * b = ab and a ⊕ b = a/b for all a, b ∈ R. (a) Does * distribute over ⊕? (b) Does ⊕ distribute over *?

Assignment

Weekend assignment

1. Two binary operations * and ⊕ are defined as m * n = mn – n – 1 and m ⊕ n = mn + n – 2 for all real numbers m and n. Find the value of 3 ⊕ (4 * 5). (a) 60 (b) 57 (c) 54 (d) 42
2. If x * y = x + y – x²y, find x when (x * 2) + (x * 3) = 63. (a) 24 (b) 22 (c) -12 (d) -21
3. A binary operation * is defined by a * b = ab. If a * b = 22 – a, find the possible values of a. (a) 1, -1 (b) 1, 2 (c) 2, -2 (d) 1, -2
4. The binary operation * is defined on the set of integers p and q by p * q = pq + p + q. Find 2 * (3 * 4). (a) 59 (b) 19 (c) 67 (d) 38
5. A binary operation ⊕ on real numbers is defined by x ⊕ y = xy + x + y for any two real numbers x and y. The value of (-3/4) ⊕ 6 is: (a) 3/4 (b) -9/2 (c) 45/4 (d) -3/4

Theory

• The operation * is defined on the set R of real numbers by a * b = (a + b)/2 – 1 for all a, b ∈ R. Is the operation * commutative in R2?
• The operation * is defined on the set R of real numbers by x * y = x + y + xx/2 for all x, y ∈ R. (a) Is the operation * commutative? (b) Is the operation * associative over the set R?

Reading assignment

• Read Binary Operation, Further Mathematics Project II, pages 13 to 22.